The key idea of the envelope method is to assume the existence of redundancy in responses that do not contribute to the estimation of \(\mathbf {\beta }\)  in model (REF ), so that the estimation of \(\mathbf {\beta }\)  is more efficient by leveraging this condition. [1]} assumes that there exists an orthogonal matrix \(({\mathbf {\Gamma }},{\mathbf {\Gamma }}_0)\in \mathbb {R}^{r\times r}\) , where \({\mathbf {\Gamma }}\in \mathbb {R}^{r\times u}\)  and \({\mathbf {\Gamma }}_0\in \mathbb {R}^{r\times (r-u)}\) , with \(0\le u\le r\)  satisfying the following conditions:
